Hyperbolic operators and Lorentzian signature lie at the heart of relativistic physics, encoding propagation, causal structure, and the distinction between initial data and subsequent evolution. Yet many approaches to quantum field theory and quantum gravity are more naturally formulated in terms of elliptic operators and Euclidean path integrals. The conventional bridge between the two is Wick rotation, in which the time coordinate is analytically continued to imaginary values. This prescription, however, hides an important structural distinction: passing from Euclidean to Lorentzian physics is not merely a change of variable, but a change in the character of the underlying differential equations and therefore in the notion of admissible data, propagation, and causality. We investigate whether this transition can instead occur dynamically. Starting from a Euclidean quantum-gravity path integral in which spacetime itself emerges from fluctuations, we ask how the inclusion of interactions can alter the principal structure of the effective theory. We identify a path for elliptic behavior to give way to hyperbolic behavior and explore the conditions under which an effective Lorentzian signature and causal structure emerge. From this perspective, Wick rotation need not be imposed as an external analytic continuation: it may arise as an effective dynamical transition from ellipticity to hyperbolicity. We will discuss connection to black hole and early universe physics.